hono: hono/zod-openapi + RPC not working

What version of Hono are you using?

Hono＠4.0.10, @hono/zod-openapi@0.9.9

What runtime/platform is your app running on?

Deno(1.41.3)

What steps can reproduce the bug?

I have prepared a minimally configured project in this repository.

https://github.com/Octo8080X/inspect-fresh-hono-openapi-rpc

Due to some issues, it appears that hono(RPC) + hono/zod-openapi is not working.

As one of the causes, there seems to be no openapi() in the object provided by the z method provided by @hono/zod-openapi.

import { Handler } from "$fresh/server.ts";
import { OpenAPIHono, createRoute, z } from "@hono/zod-openapi";

const ParamsSchema = z.object({});

console.log(z.string());
// => ZodString {
//      spa: [Function: bound safeParseAsync] AsyncFunction,
//      _def: { checks: [], typeName: "ZodString", coerce: false },
//      parse: [Function: bound parse],
//      safeParse: [Function: bound safeParse],
//      parseAsync: [Function: bound parseAsync] AsyncFunction,
//      safeParseAsync: [Function: bound safeParseAsync] AsyncFunction,
//      refine: [Function: bound ],
//      refinement: [Function: bound refinement],
//      superRefine: [Function: bound superRefine],
//      optional: [Function: bound ],
//      nullable: [Function: bound ],
//      nullish: [Function: bound nullish],
//      array: [Function: bound array],
//      promise: [Function: bound promise],
//      or: [Function: bound or],
//      and: [Function: bound and],
//      transform: [Function: bound ],
//      brand: [Function: bound brand],
//      default: [Function: bound ],
//      catch: [Function: bound catch],
//      describe: [Function: bound describe],
//      pipe: [Function: bound pipe],
//      readonly: [Function: bound readonly],
//      isNullable: [Function: bound isNullable],
//      isOptional: [Function: bound isOptional]
//    }

const HeloWorldSchema = z
  .object({
    message: z.string().openapi({  // <= Property 'openapi' does not exist on type 'ZodString'.
      example: "Hello World!",
    }),
  })
  .openapi("HeloWorldResponse");  // <= Property 'openapi' does not exist on type 'ZodObject<{ message: any; }, "strip", ZodTypeAny, { [x: string]: any; message?: any; }, { [x: string]: any; message?: any; }>'

const route = createRoute({
  method: "get",
  path: "/api/hello_world",
  request: {
    params: ParamsSchema,
  },
  responses: {
    200: {
      content: {
        "application/json": {
          schema: HeloWorldSchema,
        },
      },
      description: "Retrieve the user",
    },
  },
});

const app = new OpenAPIHono();

const appRoutes = app.openapi(route, (c) => {
  return c.json({
    message: "Hello World!",
  });
});

export type AppRoutesType = typeof appRoutes;

Again, an error is made here by creating a client using a type created from a definition that is probably faulty in some way.

import { useEffect, useState } from "preact/hooks";
import { hc } from "$hono/mod.ts";
import type { AppRoutesType } from "../routes/api/[...path].ts";
const client = hc<AppRoutesType>("http://localhost:8000/");   //  <= Type mismatch

export default function HelloWorld() {
  // omission
}

RPC with openapi removed works correctly.

What is the expected behavior?

RPC can be used without causing type errors.

What do you see instead?

No response

Additional information

Perhaps this issue illustrates the same problem. https://github.com/honojs/hono/issues/1784